A-small-ball-is-dropped-from-a-height-of-1m-into-a-horizontal-floor-Each-time-it-rebounces-to-3-5-of-the-height-it-has-fallen-a-show-that-when-the-ball-strikes-the-ground-for-the-third-time-it-has-

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Question Number 32312 by NECx last updated on 23/Mar/18

$$Asmallballisdroppedfroma$$$$heightof1mintoahorizontal$$$$floor.Eachtimeitrebouncesto$$$$3/5oftheheightithasfallen.$$$$a)showthatwhentheballstrikes$$$$thegroundforthethirdtime,it$$$$hastravelledadistanceof2.92m$$$$b)Showthatthetotaldistance$$$$travelledbytheballcantexceed$$$$4m.$$

Commented by NECx last updated on 23/Mar/18

$$pleasehelpwiththis$$

Answered by mrW2 last updated on 23/Mar/18

$$1ststrike:l_1=1$$$$2ndstrike:l_2=1+\frac{3}{5}\times 2$$$$3rdstrike:l_3=1+\frac{3}{5}\times 2+{\left(\frac{3}{5}\right)}^2\times 2=2.92m$$$$\dots .$$$$n-thstrike:l_n=1+\frac{3}{5}\times 2+{\left(\frac{3}{5}\right)}^2\times 2+\dots +{\left(\frac{3}{5}\right)}^{n-1}\times 2$$$$l_n=2[1+\frac{3}{5}+{\left(\frac{3}{5}\right)}^2+\dots +{\left(\frac{3}{5}\right)}^{n-1}]-1$$$$\underset{n\to \mathrm{\infty }}{\lim }{l}_n=\frac{2}{1-\frac{3}{5}}-1=4m$$$$\Rightarrow l_n<4m$$

Commented by NECx last updated on 25/Mar/18

$$wow\dots ..Thanks$$

Commented by NECx last updated on 25/Mar/18

$$IfImayask,whatdoesthe\times 2$$$$represents.Sincetheobjectreboundes$$$$\frac{3}{5}oftheactualheightIthought$$$$itwouldhavebeen$$$$l_2=1+\left(\frac{3}{5}\right)\times 2$$$$$$$$pleaseexplain.Thanks$$

Commented by mrW2 last updated on 06/Apr/18

$$youarerightsir.itisalsothatwhat$$$$Iwrote:$$$$l_2=1+\left(\frac{3}{5}\right)\times 2$$$$l_3=1+\left(\frac{3}{5}\right)\times 2+{\left(\frac{3}{5}\right)}^2\times 2$$$$\dots \dots$$$$l_n=1+\left(\frac{3}{5}\right)\times 2+{\left(\frac{3}{5}\right)}^2\times 2+\dots +{\left(\frac{3}{5}\right)}^{n-1}\times 2$$

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